The Door is a 1968 WB-7A cartoon. Much like Norman Normal which was released the same year, it is neither part of the Looney Tunes or Merrie Melodies series (though unlike Norman Normal it was not released as a "Cartoon Special" either).
Two stylized, watercolor-rendered American Indians go hunting, to the scat-sung, mumbled soundtrack. They come upon an Indian maiden who shows them a strange door. They open the door and see live-action shots of busy cities, war, and other modern chaos. As they see this, the music becomes more and more frantic. The ending of the cartoon shows an atomic bomb exploding as the door closes, revealing the precautionary, and very 1960s message: "peace."
It's an interesting film, but frankly, it isn't particularly enjoyable more than once or twice. Perhaps the reason it is so rarely seen, and has never aired on television alongside the other cartoons, is that it might not appeal to children.
This used to be considered by many to be one of the rarest cartoons Warner Bros. has ever made, due to the fact that it wasn't designed to interest children. It is even rare in availability on internet streamings; YouTube doesn't have it, Dailymotion does have only one copy of it but in very terrible quality.
The short was, however, released as a special feature on Disc 3 of the Looney Tunes Platinum Collection: Volume 1 Blu-Ray set in 2011. Prior to that, it was never released on home video.